A stationary AR(2) with a coefficient above one

Question

Consider the AR(2) process Xt = 1.2\,X{t-1} - 0.4\,X{t-2} + \varepsilont with white-noise shocks. Is it stationary? Answer yes or no.

Accepted Answer

It is tempting to reject this at a glance because \phi1 = 1.2 1, but for an AR(2) the first coefficient alone is not the test. Check all three conditions with \phi1 = 1.2, \phi2 = -0.4: 1. \phi1 + \phi2 = 1.2 - 0.4 = 0.8 < 1. OK. 2. \phi2 - \phi1 = -0.4 - 1.2 = -1.6 < 1. OK. 3. -0.4 < 1. OK. All three hold, so the process is stationary, despite \phi1 exceeding 1. The negative second coefficient pulls the process back and keeps the characteristic roots outside the unit circle. This is exactly why you must check the joint conditions rather than eyeballing a single coefficient. Takeaway. In an A…

A stationary AR(2) with a coefficient above one

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